Buchsbaum ring
In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence. A sequence ( a 1 , ⋯ , a r ) {\displaystyle (a_{1},\cdots ,a_{r})} of the maximal ideal m {\displaystyle m} is called a weak sequence if m ⋅ ( ( a 1 , ⋯ , a i − 1 ) : a i ) ⊂ ( a 1 , ⋯ , a i − 1 ) {\displaystyle m\cdot ((a_{1},\cdots ,a_{i-1})\colon a_{i})\subset (a_{1},\cdots ,a_{i-1})} for all i {\displaystyle i} . They were introduced by Jürgen Stückrad and Wolfgang Vogel (1973) and are named after David Buchsbaum. Every Cohen–Macaulay local ring is a Buchsbaum ring. Every Buchsbaum ring is a generalized Cohen–Macaulay ring.
In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence. A sequence of the maximal ideal is called a weak sequence if for all .
They were introduced by Jürgen Stückrad and Wolfgang Vogel (1973) and are named after David Buchsbaum.
Every Cohen–Macaulay local ring is a Buchsbaum ring. Every Buchsbaum ring is a generalized Cohen–Macaulay ring.
References
[edit]- Buchsbaum, D. (1966), "Complexes in local ring theory", in Herstein, I. N. (ed.), Some aspects of ring theory, Centro Internazionale Matematico Estivo (C.I.M.E.). II Ciclo, Varenna (Como), 23-31 agosto, vol. 1965, Rome: Edizioni cremonese, pp. 223–228, ISBN 978-3-642-11035-1, MR 0223393
- Goto, Shiro (2001) [1994], "Buchsbaum ring", Encyclopedia of Mathematics, EMS Press
- Stückrad, Jürgen; Vogel, Wolfgang (1973), "Eine Verallgemeinerung der Cohen-Macaulay Ringe und Anwendungen auf ein Problem der Multiplizitätstheorie", Journal of Mathematics of Kyoto University, 13: 513–528, ISSN 0023-608X, MR 0335504
- Stückrad, Jürgen; Vogel, Wolfgang (1986), Buchsbaum rings and applications, Berlin, New York: Springer-Verlag, ISBN 978-3-540-16844-7, MR 0881220
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